Moment map in Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $2$

In the paper Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks 2, in the part of the preliminaries the author considers a Hamiltonian action of the isometry group $$G$$ of $$\mathbb{C}\mathbb{P}^n$$ on $$\mathbb{C}\mathbb{P}^n$$. The action is described in terms of the dual of the lie algebra and then he claims that for $$\xi$$ in the lie algebra of a maximal torus $$f_{\xi}(x):=\langle \Phi(x),\xi\rangle = \frac{x^T\xi x}{2\pi i\|x\|^2}$$.Then it's claimed that we can pick a $$\xi$$ such that the flow $$\psi_t$$ of the vector field generated by it is periodic with period one , and that $$\psi_t(\mathbb{R}\mathbb{P}^n)\cap \mathbb{R}\mathbb{P}^n = \operatorname{Crit}(f_{\xi})=n+1$$.

Now I have tried to see this why this is true but I couldn't. I am only used to working with hamiltonian actions of torus $$\mathbb{T}^n$$ , where there isn't any talk about lie algebras and their dual, so I was wondering if we could describe the action of $$G$$ as an action of a torus so that I could check the statements.

Or if anyone could enlighten me why this statements are true I would appreciated it.

The choice of $$\xi$$ amounts to choosing a Hamiltonian circle action on $$\mathbb{CP}^n$$ by isometries. I.e. choosing a suitable 1-parameter subgroup of the $$PU(n+1,\mathbb{C})$$.

Consider the Hamiltonian circle action $$z.[z_{0} : z_{1} : \ldots : z_{n}] = [z_{0} : z z_{1} : \ldots :z^{n} z_{n}] ,$$

Which will have Hamiltonian $$H=f_{\xi} = \frac{\sum_{i=0}^{n} i |z_{i}|^2}{\sum_{i=0}^{n} |z_i|^2}$$. In the lie algebra of the maximal torus this is the point $$\xi = (0,1,2,\ldots,n)$$.

Then (as for any Hamiltonian circle action) the critical points are exactly the fixed points of the circle action; namely the $$n+1$$ points $$[1:0:\ldots:0]$$, $$\ldots$$, $$[0:\ldots :1]$$.

For the final claim suppose that $$[z_{0} : z z_{1} : \ldots :z^{n} z_{n}] \in \mathbb{RP}^n \;\;\forall z .$$

Clearly the $$n+1$$ fixed points above satisfy this condition, we show there are no additional ones. Suppose that more than one of the homogenous co-ordinates is $$0$$, say $$z_i$$,$$z_j$$ with $$i \neq j$$. Taking $$z=0$$ which shows that we can start with all the $$z_i$$ real, if the two co-ordinates $$z_i$$, $$z_{j}$$ are non-zero and real with $$i \neq j$$; clearly $$\frac{z^i . z_{i}}{z^j . z_j}$$ can't be real for all $$z \in S^1$$ showing that the orbit of this point is not contained in $$\mathbb{RP}^n$$.

• Thanks. And so I can use the vector generated by this aciton and it's flow to understand the rest of the paper , i.e. , it will be equivalent to the one he is refering to ? @Nick L
– user174565
Jun 24, 2021 at 16:26
• For, this I am not sure. My (perhaps naive) feeling is that any choice having isolated fixed points will satisfy the assertions you mention. I.e. if we take $\xi = (0,k_1,k_2, \ldots,, \ldots, k_{n})$ with $k_i$ strictly increasing then I think the above argument will go through, and this really will give rather different Hamiltonian circle actions (i.e. different weights at the fixed points etc). However, the choice of $\xi$ I gave in the answer is certainly the simplest one, so I think it is safe to assume they use that one. Jun 24, 2021 at 16:33